Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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19 views

Proving a formula for the coordinate representation of a mapping inbetween smooth manifolds

Let $M, N$ be smooth manifolds, and let $f: M \to N$ be a smooth mapping. I now want to prove: If $(U, \phi = (x_1, ..., x_m))$ and $(V, \psi = (y_1, ..., y_n))$ are charts for $M$ and $N$ ...
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2answers
27 views

Closed unit square is connected.

Question. Prove that the closed unit square $S=\{x$ in $\Bbb R^2 : 0 \le x_1 \le 1, 0 \le x_2 \le1 \}$ is connected. I understood how to prove unit interval is connected and I am trying to extend to ...
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0answers
41 views

Why is $\int\limits_0^1 \frac{x^{100}}{x+\alpha}dx \le \frac{1}{1+\alpha}$?

Found in my course notes: "$\int\limits_0^1 \frac{x^{100}}{x+\alpha}dx \le \frac{1}{1+\alpha}$" My conjecture: $\int\limits_0^1 \frac{x^{100}}{x+\alpha}dx = \int\limits_0^1 \frac{1}{x+\alpha}\cdot ...
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1answer
23 views

Application of Rouche theorem in order to find the roots of a polynomial in each quadrant.

I want to solve the following : (i) Show that $z^4+2z^2-z+1$ has exactly one root per plane quadrant. My idea to prove (i) is by using Rouche theorem, by considering 4 cuts of the complex plane ...
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2answers
28 views

Fundamental Solution to 2nd Order ODE

I'm currently doing a problem with the fundamental solution for $$-u''+k^2u=f(x) \quad , \quad -\infty < x < \infty$$ I'm wondering if fundamental solutions are supposed to satisfy the ...
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0answers
40 views

Application of Anova [closed]

I have a dataset for car crash data in the following format. It has 352 observations with missing values in between CarID & Year (make, model and a year of a car) – qualitative variable - ...
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1answer
11 views

Is the map $\omega\mapsto (X(\omega),Y(\omega))$ measurable with respect to $\sigma(X,Y)$?

Definitions: We have a measure space $(\Omega,\sigma(X,Y),\mu)$ where $X,Y$ are maps from $\Omega$ to some measure space $(S,\Sigma,m)$. Here $\sigma(X,Y)$ is the smallest sigma algebra that makes ...
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3answers
45 views

Homeomorphism from $(0,1)$ to $\mathbb{R}$

I want to show that $(0,1)$ is homeomorphic to $\mathbb{R}$ by finding a homeomorphism between the two. I think the function will be related to $tan(x)$ but I'm stuck on how to modify it to fit the ...
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2answers
34 views

Are these examples of a normed space and a Banach space?

Our professor gave two examples of spaces of sequences one of which is Banach and the other not: Consider the space of sequences $X$, where only finitely many terms are non-zero, with the norm ...
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1answer
46 views

Prove regarding net convergence

Let $M$ be a non-empty set and let $\mathfrak{X}$ be the set of all finite subsets of $M$. Let $\mathfrak{X}$ be a directed set by the relation $$\forall X,Y \in \mathfrak{X} \:\:\: X\le Y ...
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4answers
72 views

Where is $|xy|$ function differentiable

I'm trying to solve this problem: Let $f(x,y) = |xy|$. Find the sets of all points $(x,y) \in \Bbb R$ where $f$ is differentiable and compute the differential in those points. Can someone explain ...
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0answers
26 views

Laplace transform of a convolution-like function

Is there a way to calculate the Laplace transform of the following function? $$ \sum_{k=1}^{+\infty}f(t-(g(t)-\theta_k))h(g(t)-\theta_k), \qquad t>0. $$ Thanks in advance.
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3answers
72 views

How to prove $r(x)>p(x)$?

Given two functions $r(x)$ and $p(x)$, both of which are defined on closed interval $x\in[a,b]$. Functions $r(x)$ and $p(x)$ also satisfy the following constraints: \begin{cases} r(a)=p(a)\\ ...
6
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1answer
68 views

Find $\lim_{n \to +\infty} \int_{0}^{\infty} e^{-x} (nx - [nx]) dx $ [closed]

Find the limit $$\lim_{n \to +\infty} \int_{0}^{\infty} e^{-x} (nx - [nx]) dx $$ where $n$ is a natural number and $[nx]$ denotes the largest integer that is not greater than $nx$.
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0answers
17 views

Differential operator with absolute sign?

From Classical and Multilinear Harmonic Analysis - Schlag, Muscalu There is a operator denoted by $|D|$, where $D$ is a convenient notation for derivative operator. What does $|D|$ mean? Is it ...
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2answers
72 views

There is no 'nice' complex logarithm

Two problems are meant to establish that no 'nice' logarithm function exists for complex numbers. The first is Let $U$ be an open set in $\mathfrak{C}\setminus\{0\}$. Suppose $h:U \to \mathfrak{C}$ ...
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1answer
25 views

Alaoglu & Krein-Milman to show bounded convex weak-$*$ closed subset has extreme point

Let $X$ be a normed linear space and $K$ a bounded convex weak-$*$ closed subset of $X^{*}$. I need to show that $K$ possesses an extreme point; however, I am not entirely sure how to do this. I ...
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2answers
46 views

Find a subset of A such that its boundary does not have measure zero

Question Find a subset $A$ of $[0,1]$ such that $A=cl(intA)$ and yet $bd(A)$ does not have measure $0$. I don't know how to construct it. I think it should be closed set, cannot be empty by ...
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1answer
27 views

Laguerre polynomials and Gram Schmidt

Last two days I was trying to solve the following problem But I couldn't. It's a problem (#5.2.2) from Mathematical Methods for Physicists by George B. Arfken and Hans J. Weber, 7th Edition. I tried ...
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0answers
21 views

Principal value integral of complex exponential

I'm reading the article Brownian distance covariance and stumbled upon a equality I can't seem to derive myself. We are first presented with the following lemma: and after stating this lemma, the ...
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2answers
36 views

Can decreasing sequence of sets with $A_i$ containing infinitely less elements than $A_{i-1}$ have finite limit?

An updated question to one I just asked. Can we have a decreasing sequence of sets $A_n$ each a subset of the natural numbers with all members containing countably infinitely many elements such that ...
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0answers
25 views

A counter example and a resolution for the same question. Which is false?

Q. Let $f :\mathbb{R}^{n} \rightarrow \mathbb{R}^{m}$ be a continuous function, if $X \subset \mathbb{R}^{n}$ is bounded then it is $F(X)$. A. If $X$ is bounded then exists $\delta$ such that $X ...
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1answer
68 views

Integration of continuous function

Let $f$ be a continuous function with $f: [0,1] \to \mathbb{R}$ such that $$\int_{0}^{1}f(x)x^{n}dx = 0$$ for all non negative $n$. Prove that $f = 0$. I tried to think this problem like this: if ...
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1answer
23 views

Is every Holder-continuous function bounded? [closed]

We call a function $f:\Omega\to\mathbb R$ $\alpha$-Holder-continuous if $|f(x)-f(y)|\leq c|x-y|^\alpha$ for all $x,y\in \Omega$ and some $c>0$ and $\alpha\in(0,1]$. Let $\Omega\subset\mathbb R^n$ ...
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1answer
35 views

Lebesgue measure of subset of $\mathbb R$

Let $E$ be the subset of $\mathbb R$ and Lebesgue measure $\lambda(E)$ is positive. Is it true that $\lambda(E)= \lambda(-E)$? I tried it for interval and found true. I tried with infimum definition ...
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votes
1answer
24 views

Riemann Integrals Problem

I'm currently studying for my Analysis exam and am struggling with the following question. Give (with proof) an example of a function f : [a, b] → R, such that |f| : [a, b] → R, x → |f(x)| is ...
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2answers
36 views

Convergent Fourier series of continuous function

Let $f$ be a continuous function. It is known that its Fourier series is convergent almost everywhere to $f$ and it may fail to converge on some measure zero set. However I would like to know whether ...
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2answers
19 views

Sequence of sets with limit contains finitely many elements

Can we have a decreasing sequence of sets with all members containing countably infinitely many elements whose limit only has finitely many elements? If so, what are some examples? (Not very sure ...
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1answer
21 views

Is $f(z)=\sqrt z$ differentiable in the complex plane?

I am wondering if someone could help me with following complex analysis question: Is $f(z)=\sqrt z$ differentiable in the complex plane? I think the answer will be everywhere but for $\theta=-\pi$ ...
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1answer
19 views

Open and closed sets in a $\infty$-metric space

Denote by $\mathcal{H}$ the set of continuous maps $ h:\mathbb{R}\rightarrow \mathbb{R}$ such that $h(0)=0$. We endow $\mathcal{H}$ with the supremum metric $$ \widehat{d}(f,g)=\sup\{\vert ...
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2answers
34 views

Let $s(x)=\sum_{n=0}^\infty \frac1{(2n+1)!}x^{2n+1}$, $c(x)=\sum_{n=0}^\infty \frac1{(2n)!}x^{2n}$ prove that: $c(x)^2-s(x)^2=1$

Let $$s(x)=\sum_{n=0}^\infty \frac1{(2n+1)!}x^{2n+1}$$ $$c(x)=\sum_{n=0}^\infty \frac1{(2n)!}x^{2n}$$ prove that $$c(x)^2-s(x)^2=1$$ I know that the following series are representations of the cosh ...
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1answer
40 views

Prove $f(x) \in C^\infty$

Let $f\colon\mathbb{R}^n\to\mathbb{R}$ such that $f(x)=\exp\left(-\dfrac{1}{1-\|x\|^2}\right)$ if $\|x\|<1$ and $f(x)=0$ if $\|x\|\geq1$ How can I prove that $f$ is from $C^\infty$ class? My work ...
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2answers
25 views

$l(f,P) \leq u(f,Q)$

Let $P$ and $Q$ be two partitions of $[a,b]$ and $f$ continuous on $[a,b]$. Prove that $l(f,P) \leq u(f,Q)$, where $l(f,P)$ and $u(f,Q)$ denote the lower Darboux sum and the upper Darboux sum ...
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0answers
20 views

Rank theorem implies inverse function theorem

I am studying analysis on $\mathbb{R}^n$ and there is this question I cannot solve. Indeed it was not asked to me in any sense, but is usual to hear people saying that rank theorem is also one of the ...
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1answer
27 views

Proof that there will always be data within 1 SD

So we just started stats at school, and our teacher told us that no matter the data, no matter how distorted or weird it is, there will always be data within 1 standard deviation of the mean. Is this ...
3
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1answer
35 views

Sum of closed spaces is not closed

I recently encountered the theorem that the sum of a closed (linear) subspace with a finite dimensional subspace is closed subspace of the Banach space in which it is contained. However, this came ...
6
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1answer
173 views

Why can't we order Complex Numbers? [duplicate]

I know this may very well be a silly question. I always hear that Complex numbers cannot be ordered. But there's something I'm missing... Why can't we just compare two complex numbers $z_1,z_2$ as ...
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0answers
16 views

$Df(x_0)$ is onto. Then there is whole neighborhood of $f(x_0)$ lying in the image of $f$.

Problem says: Suppose that $f:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}$ is of class $\mathcal{C}^{1}$ and $Df(x_{0})$ has rank $m$ . This means that $Df(x_{0})$ as a linear map is ...
2
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2answers
52 views

about limit of a sequence

In investigating of convergence of a sequence we use $n\longrightarrow \infty $ . why we can only use $\infty$ and we can not use the other numbers for convergence in a sequence as convergence of the ...
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1answer
72 views

Determine whether the differential operator is compact in the following cases

Given the differential operator $\displaystyle Tx(t)=\frac{dx}{dt}$, I need to determine (and be able to justify) whether it is compact in the following three cases: $T: C^{1}[0,1]\mapsto C[0,1]$ ...
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0answers
34 views

How to prove Rouche's theorem in $\mathbb R^2$

Actually I want to prove the open mapping theorem for $\mathbb R^2$ under C-R equation. The proof of complex version is same as of $\mathbb R^2$ version before I encounter "Rouche's theorm" . So, ...
2
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1answer
37 views

Let $f$ be positive and Lebesgue measurable on $[0,1]$. Show that $\inf_{\lambda(E)\geq \epsilon} \int_E fd\lambda >0$ for any $\epsilon\in(0,1]$.

The title says it all. I've already shown, for an earlier part of this problem, that for any $E$ with $\lambda(E)>0$, we have $\int_E fd\lambda >0$. I did that by reductio, showing that ...
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1answer
55 views

Decomposition of complex Radon measures

Suppose you have a complex Radon measure $\mu$, treated as a distribution. Then does every such Radon measure admit a decomposition of the form $\mu = \sum_{n=1}^\infty c_n \delta(x-\tau_n) + \hat f$ ...
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0answers
46 views

Solving integral inequalities ( Gronwall) [closed]

I don't know how to solve this inequality $$ v'(t) \leq ct +(v(t))^p, \qquad p >1 \, \quad \mathrm{and}\ \quad , c > 0 $$ With thanks.
4
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2answers
133 views

This integral is defined ? $\displaystyle\int_0^0\frac 1x\:dx$

$\displaystyle\int_0^0\frac 1x\:dx$ This integral is defined ? If it define, what is the value ? Please,step by step
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1answer
19 views

$C(\overline\Omega)$ and $C_0(\Omega)$

Suppose $\Omega$ is a smooth open bounded domain in $\mathbb{R}^n$. Is $C_0(\Omega)$ a subset of $C(\overline\Omega)$? I think it is not because I can take $\Omega = (0,1)$ with $f(x) = \frac{1}{x}$ ...
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1answer
92 views

Proving analyticity of an integral function over $\mathbb{R}^{n}$

Let $U\subsetneqq\mathbb{R}^{n}$ be open, $\varepsilon>0$ and consider the function ...
4
votes
1answer
39 views

Show that if $ |f( \frac{1}{n}) | \leq \frac{1}{n!}$ then $0$ is an essential singularity

Given holomorphic non-constant function $f:D(0,1) \smallsetminus \{0\} \rightarrow \mathbb{C}$ so $\forall n=2,3,...:\ |f(\frac{1}{n})| \leq \frac{1}{n!}$ I need do show that $0$ is an essential ...
4
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3answers
44 views

Examples of sequences of positive terms $\{a_n\}$ such that $a_n^{1/n}\rightarrow l ~~\text{does not imply}~~ \frac{a_{n+1}}{a_n}\rightarrow l$

Give some examples of sequences of positive terms $\{a_n\}$ such that $$a_n^{1/n}\rightarrow l ~~\text{does not imply}~~ \frac{a_{n+1}}{a_n}\rightarrow l$$ If $a_n>0$ for all $n\in ...
0
votes
1answer
17 views

Limit of an absolute function

I'm currently studying for my Analysis exam, but can't seem to get the proof for the following limit. Let $f : ℝ→ ℝ$ be defined by: $$f(x) = x^2 − |x|, x ∈ ℝ$$ Using the definition of limit of a real ...